Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 70.1% → 99.8%

Time: 17.4s

Alternatives: 20

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{x + \left(y + 1\right)}}{y + x} \cdot \frac{x}{y + x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (* (/ (/ y (+ x (+ y 1.0))) (+ y x)) (/ x (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	return ((y / (x + (y + 1.0))) / (y + x)) * (x / (y + x));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((y / (x + (y + 1.0d0))) / (y + x)) * (x / (y + x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return ((y / (x + (y + 1.0))) / (y + x)) * (x / (y + x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return ((y / (x + (y + 1.0))) / (y + x)) * (x / (y + x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x)) * Float64(x / Float64(y + x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((y / (x + (y + 1.0))) / (y + x)) * (x / (y + x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+r+ 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. associate-*l/ 80.6%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. times-frac 99.8%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

   5. *-commutative 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

4. Final simplification 99.8%

   \[\leadsto \frac{\frac{y}{x + \left(y + 1\right)}}{y + x} \cdot \frac{x}{y + x} \]

Alternative 2: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+172}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -5.2e+172)
     (* t_0 (/ (/ y x) (+ y x)))
     (if (<= x -8.5e-21)
       (* (/ y (+ x (+ y 1.0))) (/ x (* (+ y x) (+ y x))))
       (* t_0 (/ (/ y (+ y 1.0)) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.2e+172) {
		tmp = t_0 * ((y / x) / (y + x));
	} else if (x <= -8.5e-21) {
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)));
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-5.2d+172)) then
        tmp = t_0 * ((y / x) / (y + x))
    else if (x <= (-8.5d-21)) then
        tmp = (y / (x + (y + 1.0d0))) * (x / ((y + x) * (y + x)))
    else
        tmp = t_0 * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.2e+172) {
		tmp = t_0 * ((y / x) / (y + x));
	} else if (x <= -8.5e-21) {
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)));
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -5.2e+172:
		tmp = t_0 * ((y / x) / (y + x))
	elif x <= -8.5e-21:
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)))
	else:
		tmp = t_0 * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -5.2e+172)
		tmp = Float64(t_0 * Float64(Float64(y / x) / Float64(y + x)));
	elseif (x <= -8.5e-21)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(t_0 * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -5.2e+172)
		tmp = t_0 * ((y / x) / (y + x));
	elseif (x <= -8.5e-21)
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)));
	else
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.2e+172], N[(t$95$0 * N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-21], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e172

   1. Initial program 62.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 80.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 80.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 80.0%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \cdot \frac{x}{x + y} \]

   if -5.2e172 < x < -8.4999999999999993e-21

   1. Initial program 67.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 90.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 90.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 90.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   if -8.4999999999999993e-21 < x

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 85.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 85.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 78.8%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around 0 84.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + y}}}{x + y} \cdot \frac{x}{x + y} \]
   5. Step-by-step derivation

      1. +-commutative 77.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 84.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{y + 1}}}{x + y} \cdot \frac{x}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 3: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+14}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-166}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -2.2e+14)
     (* t_0 (/ (/ y x) (+ y x)))
     (if (<= x -8.5e-166)
       (* (/ x (* (+ y x) (+ y x))) (/ y (+ y 1.0)))
       (/ t_0 (+ y (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -2.2e+14) {
		tmp = t_0 * ((y / x) / (y + x));
	} else if (x <= -8.5e-166) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else {
		tmp = t_0 / (y + (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-2.2d+14)) then
        tmp = t_0 * ((y / x) / (y + x))
    else if (x <= (-8.5d-166)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0d0))
    else
        tmp = t_0 / (y + (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -2.2e+14) {
		tmp = t_0 * ((y / x) / (y + x));
	} else if (x <= -8.5e-166) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else {
		tmp = t_0 / (y + (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -2.2e+14:
		tmp = t_0 * ((y / x) / (y + x))
	elif x <= -8.5e-166:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0))
	else:
		tmp = t_0 / (y + (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -2.2e+14)
		tmp = Float64(t_0 * Float64(Float64(y / x) / Float64(y + x)));
	elseif (x <= -8.5e-166)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -2.2e+14)
		tmp = t_0 * ((y / x) / (y + x));
	elseif (x <= -8.5e-166)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	else
		tmp = t_0 / (y + (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e+14], N[(t$95$0 * N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-166], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e14

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 85.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \cdot \frac{x}{x + y} \]

   if -2.2e14 < x < -8.5e-166

   1. Initial program 82.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 97.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 97.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 97.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 97.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 96.1%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 96.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 96.1%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]

   if -8.5e-166 < x

   1. Initial program 66.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 83.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 83.0%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 83.0%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 83.0%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 70.8%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 73.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 73.1%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 73.1%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 73.1%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 73.1%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 59.3%

      \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{y + \left(x + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-166}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t_0}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (/ (/ y x) t_0)))
   (if (<= x -8.8e+15)
     t_1
     (if (<= x -5.2e-25)
       (/ x (* (+ y 1.0) (+ y x)))
       (if (<= x -1e-134) t_1 (/ (/ x (+ y x)) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -8.8e+15) {
		tmp = t_1;
	} else if (x <= -5.2e-25) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = t_1;
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = (y / x) / t_0
    if (x <= (-8.8d+15)) then
        tmp = t_1
    else if (x <= (-5.2d-25)) then
        tmp = x / ((y + 1.0d0) * (y + x))
    else if (x <= (-1d-134)) then
        tmp = t_1
    else
        tmp = (x / (y + x)) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -8.8e+15) {
		tmp = t_1;
	} else if (x <= -5.2e-25) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = t_1;
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = (y / x) / t_0
	tmp = 0
	if x <= -8.8e+15:
		tmp = t_1
	elif x <= -5.2e-25:
		tmp = x / ((y + 1.0) * (y + x))
	elif x <= -1e-134:
		tmp = t_1
	else:
		tmp = (x / (y + x)) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(y / x) / t_0)
	tmp = 0.0
	if (x <= -8.8e+15)
		tmp = t_1;
	elseif (x <= -5.2e-25)
		tmp = Float64(x / Float64(Float64(y + 1.0) * Float64(y + x)));
	elseif (x <= -1e-134)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = (y / x) / t_0;
	tmp = 0.0;
	if (x <= -8.8e+15)
		tmp = t_1;
	elseif (x <= -5.2e-25)
		tmp = x / ((y + 1.0) * (y + x));
	elseif (x <= -1e-134)
		tmp = t_1;
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -8.8e+15], t$95$1, If[LessEqual[x, -5.2e-25], N[(x / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], t$95$1, N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8e15 or -5.2e-25 < x < -1.00000000000000004e-134

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 89.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 89.1%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 89.1%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 89.1%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 76.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 80.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 80.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

      2. div-inv 99.7%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x \cdot \frac{1}{\color{blue}{\frac{1}{\frac{y}{x + y}}}}}{x + y}}{y + \left(x + 1\right)} \]

      4. remove-double-div 99.9%

         \[\leadsto \frac{\frac{x \cdot \color{blue}{\frac{y}{x + y}}}{x + y}}{y + \left(x + 1\right)} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{y}{x + y}}}{x + y}}{y + \left(x + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.6%

         \[\leadsto \frac{\frac{\color{blue}{\frac{x \cdot y}{x + y}}}{x + y}}{y + \left(x + 1\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

   7. Simplified 99.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

   8. Taylor expanded in x around inf 70.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{y + \left(x + 1\right)} \]

   if -8.8e15 < x < -5.2e-25

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.7%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0 91.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + y\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   10. Simplified 91.0%

       \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 82.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 73.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 73.3%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 60.0%

      \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{y + \left(x + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \frac{\frac{y}{y + x}}{t_0}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (/ (/ y (+ y x)) t_0)))
   (if (<= x -1.25e+14)
     t_1
     (if (<= x -1.15e-26)
       (/ x (* (+ y 1.0) (+ y x)))
       (if (<= x -1e-134) t_1 (/ (/ x (+ y x)) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / (y + x)) / t_0;
	double tmp;
	if (x <= -1.25e+14) {
		tmp = t_1;
	} else if (x <= -1.15e-26) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = t_1;
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = (y / (y + x)) / t_0
    if (x <= (-1.25d+14)) then
        tmp = t_1
    else if (x <= (-1.15d-26)) then
        tmp = x / ((y + 1.0d0) * (y + x))
    else if (x <= (-1d-134)) then
        tmp = t_1
    else
        tmp = (x / (y + x)) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / (y + x)) / t_0;
	double tmp;
	if (x <= -1.25e+14) {
		tmp = t_1;
	} else if (x <= -1.15e-26) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = t_1;
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = (y / (y + x)) / t_0
	tmp = 0
	if x <= -1.25e+14:
		tmp = t_1
	elif x <= -1.15e-26:
		tmp = x / ((y + 1.0) * (y + x))
	elif x <= -1e-134:
		tmp = t_1
	else:
		tmp = (x / (y + x)) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(y / Float64(y + x)) / t_0)
	tmp = 0.0
	if (x <= -1.25e+14)
		tmp = t_1;
	elseif (x <= -1.15e-26)
		tmp = Float64(x / Float64(Float64(y + 1.0) * Float64(y + x)));
	elseif (x <= -1e-134)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = (y / (y + x)) / t_0;
	tmp = 0.0;
	if (x <= -1.25e+14)
		tmp = t_1;
	elseif (x <= -1.15e-26)
		tmp = x / ((y + 1.0) * (y + x));
	elseif (x <= -1e-134)
		tmp = t_1;
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.25e+14], t$95$1, If[LessEqual[x, -1.15e-26], N[(x / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], t$95$1, N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e14 or -1.15000000000000004e-26 < x < -1.00000000000000004e-134

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 89.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 89.1%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 89.1%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 89.1%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 76.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 80.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 80.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around inf 71.3%

      \[\leadsto \frac{\frac{\color{blue}{y}}{x + y}}{y + \left(x + 1\right)} \]

   if -1.25e14 < x < -1.15000000000000004e-26

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.7%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0 91.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + y\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   10. Simplified 91.0%

       \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 82.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 73.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 73.3%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 60.0%

      \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{y + \left(x + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ y (+ x 1.0))))
   (if (<= x -1.6e+14)
     (* t_0 (/ (/ y x) (+ y x)))
     (if (<= x -1.7e-24)
       (/ x (* (+ y 1.0) (+ y x)))
       (if (<= x -1e-134) (/ (/ y (+ y x)) t_1) (/ t_0 t_1))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (x <= -1.6e+14) {
		tmp = t_0 * ((y / x) / (y + x));
	} else if (x <= -1.7e-24) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = (y / (y + x)) / t_1;
	} else {
		tmp = t_0 / t_1;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y + x)
    t_1 = y + (x + 1.0d0)
    if (x <= (-1.6d+14)) then
        tmp = t_0 * ((y / x) / (y + x))
    else if (x <= (-1.7d-24)) then
        tmp = x / ((y + 1.0d0) * (y + x))
    else if (x <= (-1d-134)) then
        tmp = (y / (y + x)) / t_1
    else
        tmp = t_0 / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (x <= -1.6e+14) {
		tmp = t_0 * ((y / x) / (y + x));
	} else if (x <= -1.7e-24) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = (y / (y + x)) / t_1;
	} else {
		tmp = t_0 / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	t_1 = y + (x + 1.0)
	tmp = 0
	if x <= -1.6e+14:
		tmp = t_0 * ((y / x) / (y + x))
	elif x <= -1.7e-24:
		tmp = x / ((y + 1.0) * (y + x))
	elif x <= -1e-134:
		tmp = (y / (y + x)) / t_1
	else:
		tmp = t_0 / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.6e+14)
		tmp = Float64(t_0 * Float64(Float64(y / x) / Float64(y + x)));
	elseif (x <= -1.7e-24)
		tmp = Float64(x / Float64(Float64(y + 1.0) * Float64(y + x)));
	elseif (x <= -1e-134)
		tmp = Float64(Float64(y / Float64(y + x)) / t_1);
	else
		tmp = Float64(t_0 / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.6e+14)
		tmp = t_0 * ((y / x) / (y + x));
	elseif (x <= -1.7e-24)
		tmp = x / ((y + 1.0) * (y + x));
	elseif (x <= -1e-134)
		tmp = (y / (y + x)) / t_1;
	else
		tmp = t_0 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+14], N[(t$95$0 * N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-24], N[(x / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.6e14

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 85.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \cdot \frac{x}{x + y} \]

   if -1.6e14 < x < -1.69999999999999996e-24

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.7%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0 91.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + y\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   10. Simplified 91.0%

       \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   if -1.69999999999999996e-24 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 99.5%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 99.5%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 99.5%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 89.0%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 89.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 89.2%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 89.2%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 89.2%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 89.2%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around inf 46.1%

      \[\leadsto \frac{\frac{\color{blue}{y}}{x + y}}{y + \left(x + 1\right)} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 82.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 73.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 73.3%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 60.0%

      \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{y + \left(x + 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 7: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-183)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 1.85e+165)
     (/ x (* (+ y x) (+ 1.0 (+ y (* x 2.0)))))
     (* (/ x (+ y x)) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-183) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.85e+165) {
		tmp = x / ((y + x) * (1.0 + (y + (x * 2.0))));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-183) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 1.85d+165) then
        tmp = x / ((y + x) * (1.0d0 + (y + (x * 2.0d0))))
    else
        tmp = (x / (y + x)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-183) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.85e+165) {
		tmp = x / ((y + x) * (1.0 + (y + (x * 2.0))));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.2e-183:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 1.85e+165:
		tmp = x / ((y + x) * (1.0 + (y + (x * 2.0))))
	else:
		tmp = (x / (y + x)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-183)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 1.85e+165)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(1.0 + Float64(y + Float64(x * 2.0)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-183)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 1.85e+165)
		tmp = x / ((y + x) * (1.0 + (y + (x * 2.0))));
	else
		tmp = (x / (y + x)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.2e-183], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+165], N[(x / N[(N[(y + x), $MachinePrecision] * N[(1.0 + N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.2e-183

   1. Initial program 65.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 75.4%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.1%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.2%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.2%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.2%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.2%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.2%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.2%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.2%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.2%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 94.3%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 94.3%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in y around 0 55.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{x \cdot \left(1 + x\right)}{y}} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 55.0%

         \[\leadsto \color{blue}{1 \cdot \frac{x}{\frac{x \cdot \left(1 + x\right)}{y} \cdot \left(y + x\right)}} \]

      2. +-commutative 55.0%

         \[\leadsto 1 \cdot \frac{x}{\frac{x \cdot \left(1 + x\right)}{y} \cdot \color{blue}{\left(x + y\right)}} \]

      3. associate-/r* 56.3%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{x}{\frac{x \cdot \left(1 + x\right)}{y}}}{x + y}} \]

      4. associate-/r/ 57.0%

         \[\leadsto 1 \cdot \frac{\color{blue}{\frac{x}{x \cdot \left(1 + x\right)} \cdot y}}{x + y} \]

      5. *-commutative 57.0%

         \[\leadsto 1 \cdot \frac{\color{blue}{y \cdot \frac{x}{x \cdot \left(1 + x\right)}}}{x + y} \]

      6. associate-/r* 58.8%

         \[\leadsto 1 \cdot \frac{y \cdot \color{blue}{\frac{\frac{x}{x}}{1 + x}}}{x + y} \]

      7. *-inverses 58.8%

         \[\leadsto 1 \cdot \frac{y \cdot \frac{\color{blue}{1}}{1 + x}}{x + y} \]

      8. div-inv 58.8%

         \[\leadsto 1 \cdot \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      9. +-commutative 58.8%

         \[\leadsto 1 \cdot \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   10. Applied egg-rr 58.8%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{y}{x + 1}}{x + y}} \]
   11. Step-by-step derivation

       1. *-lft-identity 58.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x + y}} \]

       2. +-commutative 58.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{1 + x}}}{x + y} \]

       3. +-commutative 58.8%

          \[\leadsto \frac{\frac{y}{1 + x}}{\color{blue}{y + x}} \]

   12. Simplified 58.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{1 + x}}{y + x}} \]

   if 2.2e-183 < y < 1.85000000000000003e165

   1. Initial program 80.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 92.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 92.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 90.6%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 98.9%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 98.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 98.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 98.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 98.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 98.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 98.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 98.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 90.6%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 90.6%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in y around inf 74.6%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + \left(y + 2 \cdot x\right)\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{x}{\left(1 + \left(y + \color{blue}{x \cdot 2}\right)\right) \cdot \left(y + x\right)} \]

   10. Simplified 74.6%

       \[\leadsto \frac{x}{\color{blue}{\left(1 + \left(y + x \cdot 2\right)\right)} \cdot \left(y + x\right)} \]

   if 1.85000000000000003e165 < y

   1. Initial program 53.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 80.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 80.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 80.1%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in y around inf 88.2%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 8: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{+16}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -2.55e+16)
     (* t_0 (/ (/ y x) (+ y x)))
     (* t_0 (/ (/ y (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -2.55e+16) {
		tmp = t_0 * ((y / x) / (y + x));
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-2.55d+16)) then
        tmp = t_0 * ((y / x) / (y + x))
    else
        tmp = t_0 * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -2.55e+16) {
		tmp = t_0 * ((y / x) / (y + x));
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -2.55e+16:
		tmp = t_0 * ((y / x) / (y + x))
	else:
		tmp = t_0 * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -2.55e+16)
		tmp = Float64(t_0 * Float64(Float64(y / x) / Float64(y + x)));
	else
		tmp = Float64(t_0 * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -2.55e+16)
		tmp = t_0 * ((y / x) / (y + x));
	else
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.55e+16], N[(t$95$0 * N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55e16

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 85.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \cdot \frac{x}{x + y} \]

   if -2.55e16 < x

   1. Initial program 69.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 86.3%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 79.9%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around 0 85.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + y}}}{x + y} \cdot \frac{x}{x + y} \]
   5. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 85.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{y + 1}}}{x + y} \cdot \frac{x}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 9: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -90000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -90000000000000.0)
   (/ (/ y x) x)
   (if (<= x -5.2e-26)
     (/ x (* (+ y 1.0) (+ y x)))
     (if (<= x -1e-134) (/ y x) (/ (/ x y) (+ y (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -90000000000000.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-26) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-90000000000000.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-5.2d-26)) then
        tmp = x / ((y + 1.0d0) * (y + x))
    else if (x <= (-1d-134)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -90000000000000.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-26) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -90000000000000.0:
		tmp = (y / x) / x
	elif x <= -5.2e-26:
		tmp = x / ((y + 1.0) * (y + x))
	elif x <= -1e-134:
		tmp = y / x
	else:
		tmp = (x / y) / (y + (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -90000000000000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.2e-26)
		tmp = Float64(x / Float64(Float64(y + 1.0) * Float64(y + x)));
	elseif (x <= -1e-134)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -90000000000000.0)
		tmp = (y / x) / x;
	elseif (x <= -5.2e-26)
		tmp = x / ((y + 1.0) * (y + x));
	elseif (x <= -1e-134)
		tmp = y / x;
	else
		tmp = (x / y) / (y + (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -90000000000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.2e-26], N[(x / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9e13

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 84.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -9e13 < x < -5.2000000000000002e-26

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.7%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0 91.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + y\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   10. Simplified 91.0%

       \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   if -5.2000000000000002e-26 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 82.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 73.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 73.3%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 59.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + \left(x + 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 10: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t_0}\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (/ (/ y x) t_0)))
   (if (<= x -6.8e+14)
     t_1
     (if (<= x -2e-24)
       (/ x (* (+ y 1.0) (+ y x)))
       (if (<= x -1e-134) t_1 (/ (/ x y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -6.8e+14) {
		tmp = t_1;
	} else if (x <= -2e-24) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = (y / x) / t_0
    if (x <= (-6.8d+14)) then
        tmp = t_1
    else if (x <= (-2d-24)) then
        tmp = x / ((y + 1.0d0) * (y + x))
    else if (x <= (-1d-134)) then
        tmp = t_1
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -6.8e+14) {
		tmp = t_1;
	} else if (x <= -2e-24) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = (y / x) / t_0
	tmp = 0
	if x <= -6.8e+14:
		tmp = t_1
	elif x <= -2e-24:
		tmp = x / ((y + 1.0) * (y + x))
	elif x <= -1e-134:
		tmp = t_1
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(y / x) / t_0)
	tmp = 0.0
	if (x <= -6.8e+14)
		tmp = t_1;
	elseif (x <= -2e-24)
		tmp = Float64(x / Float64(Float64(y + 1.0) * Float64(y + x)));
	elseif (x <= -1e-134)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = (y / x) / t_0;
	tmp = 0.0;
	if (x <= -6.8e+14)
		tmp = t_1;
	elseif (x <= -2e-24)
		tmp = x / ((y + 1.0) * (y + x));
	elseif (x <= -1e-134)
		tmp = t_1;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -6.8e+14], t$95$1, If[LessEqual[x, -2e-24], N[(x / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], t$95$1, N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e14 or -1.99999999999999985e-24 < x < -1.00000000000000004e-134

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 89.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 89.1%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 89.1%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 89.1%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 76.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 80.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 80.6%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 80.6%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

      2. div-inv 99.7%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x \cdot \frac{1}{\color{blue}{\frac{1}{\frac{y}{x + y}}}}}{x + y}}{y + \left(x + 1\right)} \]

      4. remove-double-div 99.9%

         \[\leadsto \frac{\frac{x \cdot \color{blue}{\frac{y}{x + y}}}{x + y}}{y + \left(x + 1\right)} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{y}{x + y}}}{x + y}}{y + \left(x + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.6%

         \[\leadsto \frac{\frac{\color{blue}{\frac{x \cdot y}{x + y}}}{x + y}}{y + \left(x + 1\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

   7. Simplified 99.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{x + y}{y}}}}{x + y}}{y + \left(x + 1\right)} \]

   8. Taylor expanded in x around inf 70.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{y + \left(x + 1\right)} \]

   if -6.8e14 < x < -1.99999999999999985e-24

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.7%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0 91.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + y\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   10. Simplified 91.0%

       \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 82.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 73.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 73.3%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 73.3%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 59.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + \left(x + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 11: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-26} \lor \neg \left(x \leq -1 \cdot 10^{-134}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.6e+14)
   (/ (/ y x) x)
   (if (or (<= x -1.15e-26) (not (<= x -1e-134)))
     (/ x (* y (+ y 1.0)))
     (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.6e+14) {
		tmp = (y / x) / x;
	} else if ((x <= -1.15e-26) || !(x <= -1e-134)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.6d+14)) then
        tmp = (y / x) / x
    else if ((x <= (-1.15d-26)) .or. (.not. (x <= (-1d-134)))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.6e+14) {
		tmp = (y / x) / x;
	} else if ((x <= -1.15e-26) || !(x <= -1e-134)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.6e+14:
		tmp = (y / x) / x
	elif (x <= -1.15e-26) or not (x <= -1e-134):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.6e+14)
		tmp = Float64(Float64(y / x) / x);
	elseif ((x <= -1.15e-26) || !(x <= -1e-134))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.6e+14)
		tmp = (y / x) / x;
	elseif ((x <= -1.15e-26) || ~((x <= -1e-134)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.6e+14], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, -1.15e-26], N[Not[LessEqual[x, -1e-134]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.6e14

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 84.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -9.6e14 < x < -1.15000000000000004e-26 or -1.00000000000000004e-134 < x

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 84.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 84.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 84.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 84.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 60.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   if -1.15000000000000004e-26 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-26} \lor \neg \left(x \leq -1 \cdot 10^{-134}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 12: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6.7:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y x) x)))
   (if (<= y -6.8e-77)
     t_0
     (if (<= y 8.8e-146) (/ y x) (if (<= y 6.7) t_0 (* (/ 1.0 y) (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -6.8e-77) {
		tmp = t_0;
	} else if (y <= 8.8e-146) {
		tmp = y / x;
	} else if (y <= 6.7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / x) / x
    if (y <= (-6.8d-77)) then
        tmp = t_0
    else if (y <= 8.8d-146) then
        tmp = y / x
    else if (y <= 6.7d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -6.8e-77) {
		tmp = t_0;
	} else if (y <= 8.8e-146) {
		tmp = y / x;
	} else if (y <= 6.7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / x) / x
	tmp = 0
	if y <= -6.8e-77:
		tmp = t_0
	elif y <= 8.8e-146:
		tmp = y / x
	elif y <= 6.7:
		tmp = t_0
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / x) / x)
	tmp = 0.0
	if (y <= -6.8e-77)
		tmp = t_0;
	elseif (y <= 8.8e-146)
		tmp = Float64(y / x);
	elseif (y <= 6.7)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / x) / x;
	tmp = 0.0;
	if (y <= -6.8e-77)
		tmp = t_0;
	elseif (y <= 8.8e-146)
		tmp = y / x;
	elseif (y <= 6.7)
		tmp = t_0;
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -6.8e-77], t$95$0, If[LessEqual[y, 8.8e-146], N[(y / x), $MachinePrecision], If[LessEqual[y, 6.7], t$95$0, N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.79999999999999966e-77 or 8.8e-146 < y < 6.70000000000000018

   1. Initial program 79.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 94.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 94.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 93.0%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 38.5%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 38.5%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 41.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -6.79999999999999966e-77 < y < 8.8e-146

   1. Initial program 59.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 74.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 74.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 74.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 74.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 68.8%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 6.70000000000000018 < y

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-*r/ 85.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

      4. *-commutative 85.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

      5. +-commutative 85.9%

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

      6. +-commutative 85.9%

         \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

      7. associate-*r/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

      8. associate-/r* 75.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

      9. *-commutative 75.5%

         \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

      10. +-commutative 75.5%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

      11. +-commutative 75.5%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      12. associate-+l+ 75.5%

          \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 80.6%

      \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. clear-num 79.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}}} \]

      2. inv-pow 79.7%

         \[\leadsto \color{blue}{{\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{-1}} \]

      3. metadata-eval 79.7%

         \[\leadsto {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\color{blue}{\left(-1\right)}} \]

      4. sqr-pow 67.4%

         \[\leadsto \color{blue}{{\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)}} \]

      5. div-inv 67.4%

         \[\leadsto {\color{blue}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{1}{\frac{x}{x + y}}\right)}}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      6. clear-num 67.4%

         \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\frac{x + y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      7. +-commutative 67.4%

         \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      8. metadata-eval 67.4%

         \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \cdot {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      9. metadata-eval 67.4%

         \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{y + \left(x + 1\right)}{\frac{x}{x + y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      10. div-inv 67.4%

          \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5} \cdot {\color{blue}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{1}{\frac{x}{x + y}}\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. clear-num 67.4%

          \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5} \cdot {\left(\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\frac{x + y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \]

      12. +-commutative 67.4%

          \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5} \cdot {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{x}\right)}^{\left(\frac{-1}{2}\right)} \]

      13. metadata-eval 67.4%

          \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5} \cdot {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

      14. metadata-eval 67.4%

          \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5} \cdot {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{\color{blue}{-0.5}} \]

   6. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5} \cdot {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. pow-sqr 79.7%

         \[\leadsto \color{blue}{{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 79.7%

         \[\leadsto {\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}\right)}^{\color{blue}{-1}} \]

      3. unpow-1 79.7%

         \[\leadsto \color{blue}{\frac{1}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}}} \]

   8. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{1}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}}} \]

   9. Taylor expanded in y around inf 75.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{y}^{2}}{x}}} \]
   10. Step-by-step derivation

       1. unpow2 75.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot y}}{x}} \]

   11. Simplified 75.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot y}{x}}} \]
   12. Step-by-step derivation

       1. associate-/l* 76.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{y}{\frac{x}{y}}}} \]

       2. associate-/r/ 77.2%

          \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]

       3. *-commutative 77.2%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y}} \]

   13. Applied egg-rr 77.2%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6.7:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 13: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+14)
   (/ (/ y x) x)
   (if (<= x -3.3e-24)
     (/ x (* y (+ y 1.0)))
     (if (<= x -8.8e-135) (/ y x) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+14) {
		tmp = (y / x) / x;
	} else if (x <= -3.3e-24) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -8.8e-135) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.45d+14)) then
        tmp = (y / x) / x
    else if (x <= (-3.3d-24)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-8.8d-135)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+14) {
		tmp = (y / x) / x;
	} else if (x <= -3.3e-24) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -8.8e-135) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e+14:
		tmp = (y / x) / x
	elif x <= -3.3e-24:
		tmp = x / (y * (y + 1.0))
	elif x <= -8.8e-135:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+14)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.3e-24)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -8.8e-135)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45e+14)
		tmp = (y / x) / x;
	elseif (x <= -3.3e-24)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -8.8e-135)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.45e+14], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.3e-24], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.8e-135], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.45e14

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 84.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -2.45e14 < x < -3.29999999999999984e-24

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 90.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   if -3.29999999999999984e-24 < x < -8.7999999999999999e-135

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -8.7999999999999999e-135 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 59.1%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e+14)
   (/ (/ y x) x)
   (if (<= x -2.1e-23)
     (/ 1.0 (/ (* y (+ y 1.0)) x))
     (if (<= x -1e-134) (/ y x) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+14) {
		tmp = (y / x) / x;
	} else if (x <= -2.1e-23) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.7d+14)) then
        tmp = (y / x) / x
    else if (x <= (-2.1d-23)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-1d-134)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+14) {
		tmp = (y / x) / x;
	} else if (x <= -2.1e-23) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.7e+14:
		tmp = (y / x) / x
	elif x <= -2.1e-23:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -1e-134:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.7e+14)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.1e-23)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -1e-134)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.7e+14)
		tmp = (y / x) / x;
	elseif (x <= -2.1e-23)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -1e-134)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.7e+14], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.1e-23], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.7e14

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 84.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -4.7e14 < x < -2.1000000000000001e-23

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 90.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 91.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 91.0%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
   7. Step-by-step derivation

      1. clear-num 90.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + 1}{\frac{x}{y}}}} \]

      2. inv-pow 90.8%

         \[\leadsto \color{blue}{{\left(\frac{y + 1}{\frac{x}{y}}\right)}^{-1}} \]

      3. metadata-eval 90.8%

         \[\leadsto {\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\color{blue}{\left(-1\right)}} \]

      4. sqr-pow 45.7%

         \[\leadsto \color{blue}{{\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\left(\frac{-1}{2}\right)}} \]

      5. associate-/r/ 45.7%

         \[\leadsto {\color{blue}{\left(\frac{y + 1}{x} \cdot y\right)}}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      6. *-commutative 45.7%

         \[\leadsto {\color{blue}{\left(y \cdot \frac{y + 1}{x}\right)}}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      7. metadata-eval 45.7%

         \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \cdot {\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      8. metadata-eval 45.7%

         \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{y + 1}{\frac{x}{y}}\right)}^{\left(\frac{-1}{2}\right)} \]

      9. associate-/r/ 45.7%

         \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{y + 1}{x} \cdot y\right)}}^{\left(\frac{-1}{2}\right)} \]

      10. *-commutative 45.7%

          \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{-0.5} \cdot {\color{blue}{\left(y \cdot \frac{y + 1}{x}\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. metadata-eval 45.7%

          \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{-0.5} \cdot {\left(y \cdot \frac{y + 1}{x}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

      12. metadata-eval 45.7%

          \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{-0.5} \cdot {\left(y \cdot \frac{y + 1}{x}\right)}^{\color{blue}{-0.5}} \]

   8. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{y + 1}{x}\right)}^{-0.5} \cdot {\left(y \cdot \frac{y + 1}{x}\right)}^{-0.5}} \]
   9. Step-by-step derivation

      1. pow-sqr 91.0%

         \[\leadsto \color{blue}{{\left(y \cdot \frac{y + 1}{x}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 91.0%

         \[\leadsto {\left(y \cdot \frac{y + 1}{x}\right)}^{\color{blue}{-1}} \]

      3. unpow-1 91.0%

         \[\leadsto \color{blue}{\frac{1}{y \cdot \frac{y + 1}{x}}} \]

      4. +-commutative 91.0%

         \[\leadsto \frac{1}{y \cdot \frac{\color{blue}{1 + y}}{x}} \]

      5. associate-*r/ 90.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \left(1 + y\right)}{x}}} \]

      6. +-commutative 90.8%

         \[\leadsto \frac{1}{\frac{y \cdot \color{blue}{\left(y + 1\right)}}{x}} \]

   10. Simplified 90.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}} \]

   if -2.1000000000000001e-23 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 59.1%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+14)
   (/ (/ y x) x)
   (if (<= x -1.4e-26)
     (/ x (* (+ y 1.0) (+ y x)))
     (if (<= x -1e-134) (/ y x) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+14) {
		tmp = (y / x) / x;
	} else if (x <= -1.4e-26) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.6d+14)) then
        tmp = (y / x) / x
    else if (x <= (-1.4d-26)) then
        tmp = x / ((y + 1.0d0) * (y + x))
    else if (x <= (-1d-134)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+14) {
		tmp = (y / x) / x;
	} else if (x <= -1.4e-26) {
		tmp = x / ((y + 1.0) * (y + x));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.6e+14:
		tmp = (y / x) / x
	elif x <= -1.4e-26:
		tmp = x / ((y + 1.0) * (y + x))
	elif x <= -1e-134:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+14)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.4e-26)
		tmp = Float64(x / Float64(Float64(y + 1.0) * Float64(y + x)));
	elseif (x <= -1e-134)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e+14)
		tmp = (y / x) / x;
	elseif (x <= -1.4e-26)
		tmp = x / ((y + 1.0) * (y + x));
	elseif (x <= -1e-134)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.6e+14], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.4e-26], N[(x / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.6e14

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 84.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -3.6e14 < x < -1.4000000000000001e-26

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      2. clear-num 99.7%

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      3. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + y}{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}} \]

      8. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{y + x}}{y}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}\right) \cdot \left(y + x\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)} \cdot \left(y + x\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right) \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0 91.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + y\right)} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   10. Simplified 91.0%

       \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right)} \cdot \left(y + x\right)} \]

   if -1.4000000000000001e-26 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 82.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 59.1%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16: 69.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-23} \lor \neg \left(x \leq -5.5 \cdot 10^{-137}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+16)
   (/ (/ y x) x)
   (if (or (<= x -3.2e-23) (not (<= x -5.5e-137))) (/ x (* y y)) (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+16) {
		tmp = (y / x) / x;
	} else if ((x <= -3.2e-23) || !(x <= -5.5e-137)) {
		tmp = x / (y * y);
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d+16)) then
        tmp = (y / x) / x
    else if ((x <= (-3.2d-23)) .or. (.not. (x <= (-5.5d-137)))) then
        tmp = x / (y * y)
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+16) {
		tmp = (y / x) / x;
	} else if ((x <= -3.2e-23) || !(x <= -5.5e-137)) {
		tmp = x / (y * y);
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.05e+16:
		tmp = (y / x) / x
	elif (x <= -3.2e-23) or not (x <= -5.5e-137):
		tmp = x / (y * y)
	else:
		tmp = y / x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+16)
		tmp = Float64(Float64(y / x) / x);
	elseif ((x <= -3.2e-23) || !(x <= -5.5e-137))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e+16)
		tmp = (y / x) / x;
	elseif ((x <= -3.2e-23) || ~((x <= -5.5e-137)))
		tmp = x / (y * y);
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.05e+16], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, -3.2e-23], N[Not[LessEqual[x, -5.5e-137]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e16

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 83.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 84.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -1.05e16 < x < -3.19999999999999976e-23 or -5.5000000000000003e-137 < x

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 84.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 84.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 78.3%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in y around inf 46.1%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 46.1%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -3.19999999999999976e-23 < x < -5.5000000000000003e-137

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 99.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 45.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-23} \lor \neg \left(x \leq -5.5 \cdot 10^{-137}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 17: 69.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6.7:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -6.8e-77)
     t_0
     (if (<= y 1.65e-146) (/ y x) (if (<= y 6.7) t_0 (/ x (* y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -6.8e-77) {
		tmp = t_0;
	} else if (y <= 1.65e-146) {
		tmp = y / x;
	} else if (y <= 6.7) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-6.8d-77)) then
        tmp = t_0
    else if (y <= 1.65d-146) then
        tmp = y / x
    else if (y <= 6.7d0) then
        tmp = t_0
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -6.8e-77) {
		tmp = t_0;
	} else if (y <= 1.65e-146) {
		tmp = y / x;
	} else if (y <= 6.7) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -6.8e-77:
		tmp = t_0
	elif y <= 1.65e-146:
		tmp = y / x
	elif y <= 6.7:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -6.8e-77)
		tmp = t_0;
	elseif (y <= 1.65e-146)
		tmp = Float64(y / x);
	elseif (y <= 6.7)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -6.8e-77)
		tmp = t_0;
	elseif (y <= 1.65e-146)
		tmp = y / x;
	elseif (y <= 6.7)
		tmp = t_0;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e-77], t$95$0, If[LessEqual[y, 1.65e-146], N[(y / x), $MachinePrecision], If[LessEqual[y, 6.7], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.79999999999999966e-77 or 1.65e-146 < y < 6.70000000000000018

   1. Initial program 79.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 94.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 94.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 94.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 94.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 38.5%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 38.5%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   6. Simplified 38.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -6.79999999999999966e-77 < y < 1.65e-146

   1. Initial program 59.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 74.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 74.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 74.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 74.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 68.8%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 6.70000000000000018 < y

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 86.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 85.9%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in y around inf 75.6%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.6%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6.7:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 18: 58.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 4.7e-76) (/ y x) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.7e-76) {
		tmp = y / x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.7d-76) then
        tmp = y / x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.7e-76) {
		tmp = y / x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.7e-76:
		tmp = y / x
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.7e-76)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.7e-76)
		tmp = y / x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.7e-76], N[(y / x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.7000000000000002e-76

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 83.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. /-rgt-identity 83.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      4. associate-+l+ 83.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 58.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 38.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 4.7000000000000002e-76 < y

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+r+ 89.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

      3. associate-*l/ 89.0%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      4. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{x + y}} \]

   4. Taylor expanded in y around inf 59.6%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 59.6%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 19: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   3. associate-*r/ 85.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(y + x\right) + 1}} \]

   4. *-commutative 85.6%

      \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) + 1} \]

   5. +-commutative 85.6%

      \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(y + x\right) + 1} \]

   6. +-commutative 85.6%

      \[\leadsto \frac{y \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(y + x\right) + 1} \]

   7. associate-*r/ 73.9%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) + 1} \]

   8. associate-/r* 76.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{y \cdot x}{y + x}}{y + x}}}{\left(y + x\right) + 1} \]

   9. *-commutative 76.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot y}}{y + x}}{y + x}}{\left(y + x\right) + 1} \]

   10. +-commutative 76.8%

       \[\leadsto \frac{\frac{\frac{x \cdot y}{\color{blue}{x + y}}}{y + x}}{\left(y + x\right) + 1} \]

   11. +-commutative 76.8%

       \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

   12. associate-+l+ 76.8%

       \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{y + \left(x + 1\right)}} \]

3. Simplified 76.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{y + \left(x + 1\right)}} \]

4. Taylor expanded in x around 0 52.6%

   \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{y + \left(x + 1\right)} \]

5. Taylor expanded in x around inf 3.9%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 3.9%

   \[\leadsto \frac{1}{x} \]

Alternative 20: 25.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ y x))

C

assert(x < y);
double code(double x, double y) {
	return y / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y / x), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. /-rgt-identity 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

   4. associate-+l+ 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 85.6%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in y around 0 46.7%

   \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

5. Taylor expanded in x around 0 26.3%

   \[\leadsto \color{blue}{\frac{y}{x}} \]

6. Final simplification 26.3%

   \[\leadsto \frac{y}{x} \]

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))